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On-Shell Methods and Effective Field Theory

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On-Shell Methods and Effective Field Theory On-Shell Methods and Effective Field Theory by Callum R. T. Jones A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in the University of Michigan 2020 Doctoral Committee: Professor Henriette Elvang, Chair Professor Ratindranath Akhoury Associate Professor David Baker Professor Finn Larsen Assistant Professor Joshua Spitz Callum R. T. Jones [email protected] ORCID ID: 0000-0002-1325-9244 © Callum R. T. Jones 2020 ACKNOWLEDGEMENTS I would like to take this opportunity to acknowledge all of the incredible people that I been privileged to know over the past five years. My outstanding collaborators: Henriette Elvang, Steve Naculich, Marios Hadjiantonis, Shruti Paranjape, Brian McPeak, Jim Liu, Sera Cremonini and Laura Johnson. With such diverse expertise, together you have helped me learn about and contribute to areas of physics that alone would have seemed impossible. I hope that we can continue to learn and argue and discover new and amazing things for years to come. Henriette Elvang in particular will forever have my gratitude for agreeing to be my PhD advisor. As I took a chance on a new life, in a new country, you took a chance on me. More than just an outstanding researcher and mentor, the conscientiousness with which you approach all aspects of academic life has given me an unparalleled role model, the example of whom I will carry with me for the rest of my professional life. Moving to Ann Arbor, tens of thousands of kilometres from every person I have ever known and loved, was the most difficult thing I have ever experienced. It would have been unbearable I hadnt found the most wonderful group of friends. Michael Viray, Brian McPeak, Shruti Paranjape, Noah Steinberg, Joshua Foster, Steve Novakov, Brandon and Kate Berg, Joe and Alissa Kleinhenz, you will never know how much your friendship means to me. Most of all, I want to acknowledge Rachel Hyneman, I don’t have the words to express how your love and support has impacted my life over these years, so these will have to do, thank you for everything. ii TABLE OF CONTENTS Acknowledgements .................................... ii List of Figures ....................................... vii List of Tables ........................................ ix List of Appendices ..................................... x Abstract ........................................... xi Chapter 1 Introduction ...................................... 1 1.1 On-Shell Methods and Feynman Diagrams ................... 1 1.2 The Landscape of Low-Energy Effective Field Theories ............ 10 1.2.1 Bottom-up Constraints: Low-Energy Theorems ............ 15 1.2.2 Top-down Constraints: Weak Gravity Conjecture ........... 18 1.3 Overview of this Thesis ............................. 21 2 Singular Soft Limits in Gauge Theory and Gravity ................. 24 2.1 Systematics of the Soft Expansion ....................... 25 2.2 Master Equation for Singular Soft Limits .................... 26 2.3 Soft Limit Consistency Conditions ....................... 30 2.3.1 Charge Conservation and the Equivalence Principle .......... 30 2.3.2 No-Go for Massless Higher Spin .................... 33 2.4 Soft Limits and Effective Field Theory ..................... 35 2.4.1 Higher-Derivative Corrections to Soft Photon Theorem ........ 35 2.4.2 Higher-Derivative Corrections to Soft Graviton Theorem ....... 37 3 Vanishing Soft Limits and the Goldstone S-Matrix ................. 40 iii 3.1 Overview of Goldstone EFTs .......................... 40 3.1.1 Structure of the Effective Action .................... 42 3.2 Subtracted Recursion Relations ......................... 44 3.2.1 Review of Soft Subtracted Recursion Relations ............ 44 3.2.2 Validity Criterion ............................ 46 3.2.3 Non-Constructibility = Triviality .................... 49 3.2.4 Implementation of the Subtracted Recursion Relations ........ 51 3.3 Soft Bootstrap .................................. 54 3.3.1 Pure Scalar EFTs ............................ 54 3.3.2 Pure Fermion EFTs ........................... 57 3.3.3 Pure Vector EFTs ............................ 58 3.4 Soft Limits and Supersymmetry ........................ 60 3.4.1 N = 1 Supersymmetry Ward Identities ................ 60 3.4.2 Soft Limits and Supermultiplets .................... 61 3.4.3 Example: Low-Energy Theorems in Supergravity ........... 64 3.5 Supersymmetric Non-linear Sigma Model ................... 65 1 3.5.1 N = 1 CP NLSM ........................... 67 1 3.5.2 N = 2 CP NLSM ........................... 69 3.6 Galileons .................................... 76 3.6.1 Supersymmetric Galileons ....................... 78 3.6.2 Supersymmetric Galileon Bootstrap I ................. 79 3.6.3 Supersymmetric Galileon Bootstrap II ................. 82 3.6.4 Vector-Scalar Special Galileon ..................... 84 3.6.5 Higher Derivative Corrections to the Special Galileon ........ 85 3.6.6 Comparison with the Field Theory KLT Relations .......... 87 4 Born-Infeld and Electromagnetic Duality at One-Loop ............... 92 4.1 Review of Born-Infeld Electrodynamics .................... 92 4.2 Overview of Method .............................. 95 4.2.1 Generalized Unitarity and Supersymmetric Decomposition ...... 95 4.2.2 Massive Scalar Extension of Born-Infeld ............... 101 4.3 Calculating mDBI4 Tree Amplitudes ...................... 105 4.3.1 General Structure ............................ 105 4.3.2 T-Duality and Low-Energy Theorems ................. 107 iv 4.3.3 Alternative Approach to Contact Terms: Massive KLT Relations ... 112 4.4 All Multiplicity Rational One-Loop Amplitudes ................ 120 4.4.1 Diagrammatic Rules for Constructing Loop Integrands ........ 120 4.4.2 Self-Dual Sector ............................ 122 4.4.3 Next-to-Self-Dual Sector ........................ 126 4.5 Quantum Electromagnetic Duality ....................... 133 5 The Black Hole Weak Gravity Conjecture with Multiple Charges ......... 137 5.1 Multi-Charge Generalization of the Weak Gravity Conjecture ......... 137 5.2 Extremality Shift ................................ 138 5.2.1 No Correction from Three-Derivative Operators ............ 139 5.2.2 Four-Derivative Operators ....................... 140 5.2.3 Leading Shift to Extremality Bound .................. 143 5.3 Black Hole Decay and the Weak Gravity Conjecture .............. 144 5.3.1 Examples ................................ 146 5.3.2 Unitarity and Causality ......................... 149 5.4 Renormalization of Four-Derivative Operators ................. 153 5.4.1 Non-Renormalization and Electromagnetic Duality .......... 153 5.4.2 RG Flow and the Multi-Charge Weak Gravity Conjecture ....... 159 6 Higher-Derivative Corrections to Entropy and the Weak Gravity Conjecture in Anti-de Sitter Space .................................. 162 6.1 Weak Gravity Conjecture and Black Hole Entropy ............... 162 6.2 Corrections to the Geometry .......................... 163 6.2.1 The Zeroth Order Solution ....................... 164 6.2.2 The First Order Solution ........................ 164 6.2.3 Asymptotic Conditions and Conserved Quantities ........... 166 6.3 Mass, Charge, and Entropy from the geometry shift .............. 169 6.3.1 Mass, Charge, and Extremality ..................... 169 6.3.2 Wald Entropy .............................. 173 6.3.3 Explicit Results for the Entropy Shifts ................. 175 6.4 Thermodynamics from the On-Shell Euclidean Action ............. 176 6.4.1 Two-Derivative Thermodynamics ................... 179 6.4.2 Four-Derivative Corrections to Thermodynamics ........... 181 6.5 Constraints From Positivity of the Entropy Shift ................ 184 v 6.5.1 Thermodynamic Stability ....................... 186 6.5.2 Constraints on the EFT Coefficients .................. 188 6.5.3 Flat Space Limit ............................ 190 6.6 Holography and the Shear Viscosity to Entropy Ratio ............. 191 6.7 Weak Gravity Conjecture in AdS ........................ 193 Appendices ......................................... 196 Bibliography ........................................ 238 vi LIST OF FIGURES FIGURE 1.1 (Left): a representation of a Wilsonian UV complete quantum field theory as an RG flow from a UV CFT to an IR CFT. (Right): many models may flow to the same IR Gaussian fixed point defined by a collection of non-interacting massless degrees-of- freedom. The low-energy EFT of these massless modes captures universal aspects of this class of models. ................................ 13 4.1 Some key-properties of BI amplitudes at tree-level, in particular the double-copy construction and 4d electromagnetic duality. The idea behind the T-duality constraint [1] is that when dimensionally reduced along one direction, a linear combination of the photon polarizations become a scalar modulus of the compactified direction,. i.e. it is the Goldstone mode of the spontaneously broken translational symmetry and as such it must have enhanced O(p2) soft behavior. ................. 93 5.1 (Left): an extremality curve that naively violates the WGC as it does not enclose the unit circle. (Right): the convex completion of the extremality curve does enclose the unit circle, hence the WGC is satisfied. For this to be possible the extremality surface must be somewhere locally non-convex, which is shown in Appendix J to be impossible in the perturbative regime. ........................ 145 5.2 (Left): the corrections to the extremality curve are everywhere positive, hence the WGC is satisfied. (Right): the
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